We define a class of measures having the following properties:
(1) the measures are supported on self-similar fractal subsets of the unit cube IM=[0,1)M, with 0 and 1 identified as necessary;
(2) the measures are singular with respect to normalized Lebesgue measure m on IM;
(3) the measures have the convolution property that μ∗Lp⊆Lp+ε for some ε = ε(p) > 0 and all p ∈ (1,∞).
We will show that if (1/p,1/q) lies in the triangle with vertices (0,0), (1,1) and (1/2,1/3), then μ∗Lp⊆Lq for any measure μ in our class.

For 1 ≤ p,q ≤ ∞, we prove that the convolution operator generated by the Cantor-Lebesgue measure on the circle is a contraction whenever it is bounded from Lp() to Lq(). We also give a condition on p which is necessary if this operator maps Lp() into L²().

A convolution operator, bounded on Lq⁢(ℝn), is bounded on Lp⁢(ℝn), with the same operator norm, if p and q are conjugate exponents. It is well known that this fact is false if we replace ℝn with a general non-commutative locally compact group G. In this paper we give a simple construction of a convolution operator on a suitable compact group G, wich is bounded on Lq⁢(G) for every q∈[2,∞) and is unbounded on Lp⁢(G) if p∈[1,2).